Local gauge symmetries Lagrangians and equations of motion

  1. Hey gang,

    I'm re-working my way through gauge theory, and I've what may be a silly question.

    Promotion of global to local symmetries in order to 'reveal' gauge fields (i.e. local phase invariance + Dirac equation -> EM gauge field) is, as far as i can tell, always done on the Lagrangian, as the Lagrangian must be invariant under the symmetry imposed.

    My question is: does this necessarily imply that the equation of motion is always invariant? If so, is the procedure of trying to find an invariant equation of motion equivalent? Is it just that due to the churning of the Lagrangian through the Euler Lagrange equations, the EoM is usually more complicated and so harder to see easy ways to make things invariant under certain operations?

  2. jcsd
  3. ChrisVer

    ChrisVer 2,403
    Gold Member

    What do you mean about the equation of motion being invariant?
    for example a non-interacting spin-1/2 field EOM is just the simple Dirac Equation.
    If you allow for the existence of the spin-1 "photonic" field, coming from the local U(1) gauge symmetry, then the Dirac equation changes (you apply the minimal coupling [itex]p^{\mu} \rightarrow p^{\mu} - q A^{\mu} [/itex]). In that sense, since your Lagrangian is invariant, then the EOM are also going to remain invariant (however they won't be the same for the 2 cases I mentioned). In the last case, both a transformed and not transformed Lagrangian are the same.
    In addition invariances (in general) are mainly to keep the action invariant and not the Lagrangian (for example the last can change up to a total derivative, and yet yield the same EoM). Now if the Lagrangian happens to remain invariant, so does the action.
  4. Yes, I guess what I am asking is does an invariant Lagrangian imply an invariant EoM? If so why not apply the gauge symmetry to the EoM and search for a wavefunction that makes the EoM invariant under the symmetry?
  5. ChrisVer

    ChrisVer 2,403
    Gold Member

    This question could also be asked in the classical mechanics....
    What's the difference between working in the Lagrangian formalism and the 2nd Law of Newton (the EoM).
    I think it's always easier to see the symmetries of the Lagrangian rather than the EoM.
  6. What do you mean by 'see' the symmetries, in this case the symmetry is imposed is it not? Do you mean see how to modify the lagrangian to make it invariant under the symmetry?
  7. ChrisVer

    ChrisVer 2,403
    Gold Member

    yes and not only.
  8. Great, thanks!
  9. ChrisVer

    ChrisVer 2,403
    Gold Member

    For example, at least as far as I've seen it, in the EoM you consider you take into account the electromagnetic interactions and thus you can apply the minimal coupling procedure... however in the Lagrangian you don't have to think of that. You just have to try turn a global symmetry that already exists into a local one, and the "electromagnetic" field appears as the connection into the covariant derivative. Also, in addition to that, the Lagrangian gives you the Maxwell equations, through the strength field tensor.
    And I guess it's even more difficult to work with other symmetries (as for example SU(2) or SU(3)) from the equations of motion (I think you can't know the last).
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